scipy.special.lpmn

scipy.special.lpmn(m, n, z)

Sequence of associated Legendre functions of the first kind.

Computes the associated Legendre function of the first kind of order m and degree n, Pmn(z) = \(P_n^m(z)\), and its derivative, Pmn'(z). Returns two arrays of size (m+1, n+1) containing Pmn(z) and Pmn'(z) for all orders from 0..m and degrees from 0..n.

This function takes a real argument z. For complex arguments z use clpmn instead.

Parameters

mint

|m| <= n; the order of the Legendre function.

nint

where n >= 0; the degree of the Legendre function. Often called l (lower case L) in descriptions of the associated Legendre function

zfloat

Input value.

Returns

Pmn_z(m+1, n+1) array

Values for all orders 0..m and degrees 0..n

Pmn_d_z(m+1, n+1) array

Derivatives for all orders 0..m and degrees 0..n

See also

clpmn

associated Legendre functions of the first kind for complex z

Notes

In the interval (-1, 1), Ferrer’s function of the first kind is returned. The phase convention used for the intervals (1, inf) and (-inf, -1) is such that the result is always real.

References

1

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

2

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3